Forcing roman domination in circulant graphs and grid graphs

A set [Formula: see text] of vertices is a dominating set if every vertex in [Formula: see text] has a neighbor in [Formula: see text]. A Roman dominating function (RDF) on a graph [Formula: see text] is defined to be a function [Formula: see text] satisfying the condition that every vertex [Formula: see text] for which [Formula: see text] is adjacent to at least one vertex [Formula: see text] for which [Formula: see text]. An RDF [Formula: see text] of [Formula: see text] can also be represented by a set of ordered pairs [Formula: see text]. A subset [Formula: see text] of [Formula: see text] is called a forcing subset of [Formula: see text] if [Formula: see text] is the unique extension of [Formula: see text] to a [Formula: see text]-function. We define a forcing Roman domination number of [Formula: see text] denoted by [Formula: see text], as [Formula: see text]is a forcing subset of[Formula: see text][Formula: see text]. The forcing Roman domination number [Formula: see text] of [Formula: see text] is defined as [Formula: see text]. Hence for every graph [Formula: see text], [Formula: see text]. In this paper, we determine the forcing Roman domination number for circulant and grid graphs.